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Statement

Setting (p. 1). $\Phi$ is the permutation of the 2-adic integers $\mathbb Z_2$ matching the expansions $Q=2^{d_0}+2^{d_1}+\cdots$ and $N=\frac{-1}{3}2^{d_0}+\frac{-1}{9}2^{d_1}+\frac{-1}{27}2^{d_2}+\cdots$ over a common increasing sequence $0\le d_0<d_1<\cdots$, with $N=\Phi(Q)$; see the conjecture's page for the full definition.

Maps (p. 2). $H(Q)=Q/2$ if $Q$ is even and $H(Q)=Q-1$ otherwise; $C(N)=N/2$ if $N$ is even and $C(N)=3N+1$ otherwise. Both act on $\mathbb Z_2$, parity being that of the 2-adic integer.

Theorem 1 (p. 2): "$C(\Phi(Q)) = \Phi(H(Q))$."

The print states the identity with no quantifier; its proof treats an arbitrary $Q\in\mathbb Z_2$, so the identity holds for every 2-adic integer $Q$. The paper notes (p. 2) that this conjugacy is equivalent to Theorem 1 of Akin's unpublished manuscript $3x+1$, and that $\Phi$ is exactly the inverse of the map $Q_\infty$ of Lagarias's 1985 survey.

Proof pointer

Direct computation from the expansions (p. 2). If $Q$ is even, every exponent in $d$ is positive (or $d$ is empty), and halving the expansion (2) of $\Phi(Q)$ lowers every exponent by one, which is the expansion of $\Phi(Q/2)$. If $Q$ is odd, $d_0=0$, and $3\Phi(Q)+1$ cancels the leading term and shifts the coefficients $-1/3^{i+1}$ down one place, giving the expansion of $\Phi(Q-1)$.

Read depth

Claims checked: the statement and the definitions of $H$ and $C$ were read on the page images of the print, and the proof was followed. A second reader checked the statement, hypotheses, label and page against the print; the proof was not independently reviewed.

Dependencies

The definition of $\Phi$ (p. 1).

Source. Daniel J. Bernstein, A non-iterative 2-adic statement of the $3N+1$ conjecture, Proceedings of the American Mathematical Society 121 (1994), 405--408. Pages are those of the author's typescript named on the source card, numbered 1--4 rather than by the journal's pagination.

Bears on

• Problem 1135: the identity turns iteration of the Collatz map $C$ into iteration of $H$ under the change of variable $\Phi$, and is the step behind Theorem 2 and Theorem 3. It says nothing by itself about whether orbits reach $1$.